When all elements in a arithmetic progression are divided by a constant number k, and are written down the remainder, you'll quickly notice that a series of numbers will appear.
simple example
7 42 77 112 147 182 217 252 287 322 357 392 (progression, 35 added each time)
7 42 17 52 27 2 37 12 47 22 57 32 (remainder after division by 60)
When 35 is added again, that gives 392+35=427. The remainder after division by 60 is again 7. The pattern
7 42 17 52 27 2 37 12 47 22 57 32
repeats itself. In this example, this pattern consists of 12 numbers.
Can you calculate the length of that pattern with a formula when you know the constant difference in the progression and the constant number? I tried to figure this out, but failed.
Answer
The period in the example is the smallest number $k$ with the property $$35k\equiv 0 \mod 60$$
We get $$k=\frac{60}{\gcd(35,60)}=\frac{60}{5}=12$$
This way you can find the period in general.
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